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(D) Consider a spherical shell of radius $x$ and thickness $dx$ between the two spheres.The area of this shell is $A = 4\pi x^2$.The resistance $dR$ o

Vedclass · Accepted Answer

$\frac{\rho}{4\pi} \left( \frac{1}{a} - \frac{1}{b} \right)$

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(D) Consider a spherical shell of radius $x$ and thickness $dx$ between the two spheres.The area of this shell is $A = 4\pi x^2$.The resistance $dR$ of this thin shell is given by $dR = \rho \frac{dx}{A} = \rho \frac{dx}{4\pi x^2}$.To find the total resistance $R$ between the spheres,we integrate from $r = a$ to $r = b$:$R = \int_a^b \frac{\rho}{4\pi x^2} dx$.$R = \frac{\rho}{4\pi} \int_a^b x^{-2} dx$.$R = \frac{\rho}{4\pi} \left[ -\frac{1}{x} \right]_a^b$.$R = \frac{\rho}{4\pi} \left( -\frac{1}{b} - (-\frac{1}{a}) \right) = \frac{\rho}{4\pi} \left( \frac{1}{a} - \frac{1}{b} \right)$.

Space between two concentric conducting spheres of radii $a$ and $b$ $(b > a)$ is filled with a medium of resistivity $\rho$. The resistance between the two spheres will be

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